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The history of a general criterium on spaceability

  • Víctor M. Sánchez EMAIL logo
Published/Copyright: March 20, 2017
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Open Mathematics
From the journal Open Mathematics Volume 15 Issue 1

Abstract

There are just a few general criteria on spaceability. This survey paper is the history of one of the first ones. Let I1 and I2 be arbitrary operator ideals and E and F be Banach spaces. The spaceability of the set of operators I1(E, F)\ I2(E, F) is studied. Before stating the criterium, the paper summarizes the main results about lineability and spaceability of differences between particular operator ideals obtained in recent years. They are the seed of the ideas contained in the general criterium.

Keywords: Spaceability; Operator ideal
MSC 2010: 47L20

1 Introduction

During the last two decades many authors have been searching for large linear structures of mathematical objects enjoying certain special properties. If X is a topological vector space, a subset A of X is said to be spaceable if A ∪ {0} contains an infinite-dimensional closed subspace. The subset A is called lineable if A ∪ {0} contains an infinite-dimensional subspace (not necessarily closed). After these definitions were introduced by Aron, Gurariy and Seoane-Sepúlveda in [1], many authors showed that many non-linear special sets can often have these properties. See, for instance, the recent works [2–8], just to cite some. However, before the publication of [1] some authors already found large linear structures enjoying special properties even though they did not explicitly used terms like lineability or spaceability (see [9, 10]). We refer the interested reader to a very recent book on the topic ([11]) where many examples can be found and techniques are developed in several different frameworks.

Surprisingly, there exist very few general criteria on spaceability. In fact, most results on concrete properties have been proved directly and constructively. Perhaps, the first general criterium on spaceability appeared in [12] where Wilansky proved that if Y is a closed vector subspace of a Banach space X, then X \ Y is spaceable if and only if Y has infinite codimension. An improvement of this result, where X is a Fréchet space, is ascribed by Kitson and Timoney to Kalton (see [13, Theorem 2.2]). Kitson and Timoney used it to obtain the following theorem:

Theorem 1.1

([13]). Let (En)n ∊ ℕ be a sequence of Banach spaces, F a Fréchet space and Tn : En ⟶ F bounded linear operators. Let G be the span of ⋃n=1∞Tn(En). If G is not closed in F, then the set F \ G is spaceable.

Three years before, lineability of the difference between a couple of particular operator ideals was studied for the first time. Thus, in [14], Puglisi and Seoane-Sepúlveda showed that if E and F are Banach spaces where E has the two series property, then the set L(E, F*)\ Π1(E, F*) is lineable, where Π1 denotes the ideal of 1-summing operators.

The following year, and partially answering a question posed in [14], Botelho, Diniz and Pellegrino proved in [15] that if E is a superreflexive Banach space containing a complemented infinite-dimensional subspace with unconditional basis, or F is a Banach space having an infinite unconditional basic sequence, then the set K(E, F)\ Πp (E, F) is lineable for every p ≥ 1, where K denotes the ideal of all compact operators.

More generally, in [13], the spaceability of the set of operators K(E, F)\ ⋃p ≥ 1 Πp(E, F) was obtained as a consequence of Theorem 1.1 when E is a superreflexive Banach space.

Finally, and recently, in [16], the authors continued this research considering general operator ideals in the sense of Pietsch I1 and I2 and Banach spaces E and F, such that the set I1(E, F)\ I2(E, F) is non-empty. They introduced the new notion of σ-reproducible space for obtaining the spaceability of this set when E or F belongs to this class of spaces and I1(E, F) is complete with the usual operator norm or with another complete ideal norm. Many classical Banach spaces are σ-reproducible, in particular, rearrangement invariant spaces and spaces of continuous functions.

For completeness, most of the original proofs have been included.

2 Preliminaries

An operator ideal in the sense of Pietsch (see [17, 18]) is defined as follows:

Definition 2.1

Let 𝔅 denote the class ofall Banach spaces and let L denote the class of all bounded linear operators between Banach spaces. An operator ideal I is a “mapping” I : 𝔅 × 𝔅 ⟶ 2L such that

  1. For each pair of Banach spaces E and F the collection of operators I(E, F) (or I(E) if E = F) is a subspace of the space L(E, F) (or L(E) if E = F) of bounded linear operators from E to F containing all finite-rank operators.

  2. If in a scheme of bounded linear operators E 0 → S 1 E → T F → S 2 F 0 we have T ∈ I(E, F), then S2 ∘ T ∘ S1∈ I(E0, F0).

Definition 2.2

An ideal norm defined on an ideal I is a rule ‖ ⋅ ‖I that assigns to every operator T ∈ I a non-negative number ‖ T ‖I satisfying the following conditions:

  1. ‖ x* ⊗ y ‖I = ‖ x*‖E*‖ y ‖F for x*∈ E*, y∈ F where (x* ⊗ y)(x) = x*(x)y for x ∈ E.

  2. ‖ S + T ‖I ≤ ‖ S ‖I + ‖ T ‖I for S, T ∈ I(E, F).

  3. ‖ S2 ∘ T ∘ S1‖I≤ ‖ S2 ‖‖ T ‖I ‖ S1‖ for S2 ∈ L(F, F0), T ∈ I(E, F) and S1∈ L(E0, E).

The last condition implies that ‖ λ T ‖I = |λ|‖ T ‖I for T ∈ I and λ ∈ 𝕂. Thus, we have indeed a norm. Moreover, ‖ T ‖≤‖ T ‖I where ‖ ⋅ ‖ denotes the usual operator norm of L(E, F), which is an example of ideal norm.

A classical closed operator ideal endowed with the canonical operator norm of L(E, F) is the ideal K of all compact operators.

A classical non-closed operator ideal with respect to the operator norm is the ideal Πq, p of (q, p)-summing operators. Recall that if 1 ≤ p ≤q < ∞, an operator T ∈ L(E, F) is called (q, p)-summing (or p-summing if p = q) if there is a constant C so that, for every choice of an integer n and vectors (xi)i=1ninE, we have

∑i=1n∥T(xi)∥q1/q≤Csup||x∗||≤1∑i=1n|x∗(xi)|p1/p.

The smallest possible constant C defines a complete ideal norm on this operator ideal, denoted by πq, p(⋅). If 1/p− 1/q ≤ 1/r−1/s and p ≤r, one has Πq, p⊂Πs, r (see [17, page 459]).

Let us recall now the definition of a rearrangement invariant space. Given a measure space (Ω, λ), where Ω is the interval [0, 1] or [0, ∞) and λ is the Lebesgue measure, or Ω = ℕ and λ is the counting measure, the distribution function λx associated to a scalar measurable function x on Ω is defined by λx(s) = λ{t − Ω : |x(t)|>s}. The decreasing rearrangement function x* of x is defined by x*(t) = inf{s ∈ [0, ∞):λx(s) ≤ t}. A Banach space (E, ‖ ⋅ ‖E) of measurable functions defined on Ω is said to be a rearrangement invariant space if the following conditions are satisfied:

  1. If y ∈ E and | x (t) |≤| y(t)|λ-a.e. on Ω, then x∈ E and ‖ x‖E≤‖ y‖E.

  2. If y∈ E and λx = λy, then x ∈ E and ‖ x ‖E = ‖ y ‖E.

Important examples of rearrangement invariant spaces are Lp, Lorentz, Marcinkiewicz and Orlicz spaces. For properties of rearrangement invariant spaces we refer to [19–21].

3 Results

The first result about lineability of the difference between a couple of particular operator ideals appeared in [14]. In this paper the authors showed that if E and F are Banach spaces where E has the two series property, then L(E, F*)\ Π1(E, F*) is lineable, where Π1 denotes the ideal of 1-summing operators.

Definition 3.1

A Banach space E is said to have the two series property provided there exist unconditionally convergent series ∑i=1∞fiinE∗and∑i=1∞xiinE such that

∑i=1∞∑j=1∞|fj(xi)|2∥fj||1/2=∞.

For instance, every Lp-space has the two series property for 1 < p < ∞.

Lemma 3.2

([14]). Let E be a Banach space satisfying the two series property. Let ∑i=1∞fiinE∗and∑i=1∞xi in E such that ∑i=1∞(∑j=1∞|fj⋅(xi)|2||fj||)1/2=∞. Then, there exists a sequence of countable pairwise disjoint subsets of ℕ, (An)n ∈ ℕ, such that

∑i=1∞∑j∈An|fj(xi)|2∥fj||1/2=∞

for each n ∈ ℕ.

Theorem 3.3

([14]). Let E be a Banach space satisfying the two series property. Then the set L(E, ℓ2)\ Π1(E, ℓ2) is lineable.

Proof

Since E satisfies the two series property, there exist unconditionally convergent series ∑i=1∞fiinE∗and∑i=1∞xi in E such that

∑i=1∞∑j=1∞|fj(xi)|2∥fj||1/2=∞.

Let (An)n ∈ ℕ be the sequence given by Lemma 3.2. For each n ∈ ℕ, let us define the operator Tn:E⟶ℓ2 by

Tn(x)=∑k∈Anfk(x)∥fk∥1/2ek.

If ‖ x ‖E ≤1, then

∥Tn(x)∥=∑k∈An|fk(x)|2||fk∥1/2≤∑k=1∞|fk(x)|2||fk∥1/2≤sup||x||=1∑k=1∞|fk(x)|1/2<∞.

Thus, Tn is well-defined and Tn∈ L(E, ℓ2) for every n ∈ ℕ.

But ∑i=1∞∥Tn(xi)∥=∞ for every n∈ ℕ. Then Tn∉ Π1(E, ℓ2) for each n∈ ℕ.

Because of the pairwise disjointness of the sets An, we have that the sequence (Tn)n∈ ℕ is linearly independent in L(E, ℓ2).

Finally, le us show that every nonzero bounded linear operator in the linear span of the sequence (Tn)n∈ ℕ does not belong to the ideal of 1-summing operators. It is enough to consider the linear combination of two elements because the general case follows similarly. Thus, if λ1, λ2∈ 𝕂 and n1, n2∈ ℕ, then (assuming, without loss of generality, that λ1≠ 0) we have that

∑i=1∞λ1Tn1(xiλ1)+λ2Tn2(xiλ1)=∑i=1∞∑k∈An1|fk(xi)|2∥fk||+∑k∈An2fk(λ2λ1xi)2∥fk∥1/2=∞.

Since ∑i=1∞xiλ1 is an unconditional convergent series in E, we obtain the result. □

Now, given two Banach spaces E and F, where E enjoys the two series property, a linearly independent sequence (Tn~)n∈N can be constructed from the sequence (Tn)n ∈ ℕ made in Theorem 3.3 whose linear span is contained in L(E, F*)\ Π1(E, F*) . Thus, we have

Corollary 3.4

([14]). Let E, F be Banach spaces, where E has the two series property. Then the set L(E, F*)\ Π1(E, F*) is lineable.

In [22], Davis and Johnson proved that the set K(E, F)\ Πp(E, F) is non-empty whenever E is a superreflexive Banach space and F is any Banach space. The question about the lineability of this set was posed in [14] and partially answered in [15, Theorem 2.1]:

Theorem 3.5

([15]). Let E be a superreflexive Banach space. If either E contains a complemented infinite- dimensional subspace with unconditional basis or F is a Banach space having an infinite unconditional basic sequence, then K(E, F)\ Πp(E, F) is lineable for every p ≥ 1.

Proof

First, let us suppose that E contains a complemented infinite-dimensional subspace E0 with unconditional basis (en)n∈ ℕ. We consider a decomposition of ℕ into infinitely many infinite pairwise disjoint subsets (Ak)k∈ ℕ. Since (en)n∈ ℕ is an unconditional basis, it is well known that (en)n∈ Ak is an unconditional basic sequence for every k ∈ ℕ. Let us denote by Ek the closed span of (en)n∈ Ak. As a subspace of a superreflexive space, Ek is superreflexive as well, and there exists Tk ∈ K(Ek, F)\ Πp(Ek, F) for each k ∈ ℕ.

If C is the unconditional basis constant of (en)n∈ ℕ, then

∑n=1∞ϵnanen≤C∑n=1∞anen

for every εn = ± 1 and scalars an. We denote by Pk : E0 ⟶ Ek the canonical projection onto Ek for each k ∈ ℕ. For x=∑n=1∞anen∈E0 we have

2Pk(x)=∑n∈Ak2anen=∑n=1∞ϵnanen+∑n=1∞ϵn′anen

for a suitable choice of signs εn and ϵn′. Thus,

∥Pk(x)∥≤C∥x∥.

So each Pk is continuous, ‖ Pk ‖≤C and Ek is complemented in E0.

If π:E ⟶ E0 is the projection onto E0, for each k∈ ℕ we define Tk~=Tk∘Pk∘π. Since (Pk ∘π)(x) = x for every x ∈ Ek, then Tk~∈K(E,F)∖Πp(E,F). Given scalars λ1,…, λn with at least one λk ≠ 0, there is a weakly p-summable sequence (xj)j∈ ℕ ⊂ Ek such that ∑j=1∞∥Tk(xj)∥p=+∞. The sequence (xj)j∈ ℕ is weakly p-summable in E, Tk~ (xj) = Tk(xj) and Ti~ (xj) = 0 if i ≠ k for every j ∈ ℕ. Thus,

∑j=1∞∥λ1T1~(xj)+⋯+λnTn~(xj)∥p=∑j=1∞∥λkTk(xj)∥p=+∞,

proving that the span of (Tk~)k∈N is contained in K(E, F)\ Πp(E, F).

Let us prove now that the set of operators (Tk~)k∈N is linearly independent. Let λ1,…, λn be scalars such that λ1T1~+⋯+λnTn~=0. Choosing xk ∈ Ek such that Tk~ (xk)≠ 0 for each k ∈ {1,…, n}, we have that

0=λ1T1~(xk)+⋯+λnTn~(xk)=λkTk~(xk).

It follows that λk = 0 for every k ∈ {1, …, n}.

Now, let us suppose that F contains a subspace F0 with unconditional basis (en)n∈ ℕ having unconditional basis constant C. Considering again the subsets (Ak)k∈ ℕ as above, we define Fk as the closed span of (en)n∈ Ak and let Pk : F0 ⟶ Fk be the corresponding projections. We also obtain as above that ‖ Pk ‖≤C. For each k∈ ℕ there exists Tk ∈ K(E, Fk) \ Πp(E, Fk).

If yi ∈ Fi and yj ∈ Fj with i ≠ j, then

∥yi∥=∥Pi(yi+yj)≤C∥yi+yj∥.

We define now the operator Tk~ by the composition of Tk with the inclusion from Fk to F. It is clear that Tk~ ∈ K(E, F)\ Πp(E, F) for each k ∈ ℕ. Since

∥Ti~(x)+Tj~(x)∥≥C−1∥Ti~(x)∥

for every x ∈ E, then Ti~+Tj~∈K(E,F)∖Πp(E,F) for all i, j ∈ ℕ. It is easy to deduce that the span of (Tk~)k∈N is contained in K(E, F)\ Πp(E, F). The linear independence is obtained as in the first case.

In [13], Theorem 3.5 was improved by establishing spaceability, and indeed a single infinite dimensional closed subspace valid for all p ≥ 1. In order to obtain this improvement we will also need the following result:

Proposition 3.6

([13]). Let En(n∈ ℕ) and F be Fréchet spaces and Tn : En ⟶ F bounded linear operators. Let G be the span of ⋃n=1∞Tn(En). If G is closed in F, then there exists k ∈ ℕ such that G is the span of ⋃n=1kTn(En).

Theorem 3.7

([13]). Let E and F be infinite dimensional Banach spaces. If E is superreflexive, then

K(E,F)∖⋃1≤p<∞Πp(E,F)

is spaceable.

Proof

Since K(E, F) is closed in the operator norm, it follows that Πp(E, F) ∩ K(E, F) is closed in Πp(E, F), hence a Banach space in the norm πp(⋅) for every p ≥ 1.

In the proof of [22, Theorem], it is shown that the norm induced by Πp(E, F) on the finite-rank operators is not equivalent to the operator norm for 1 ≤ p < ∞ when E is superreflexive. Thus, Πp(E, F)∩ K(E, F) is not closed in K(E, F).

Due to Proposition 3.6 the union

⋃1≤p<∞Πp(E,F)∩K(E,F)=⋃p∈NΠp(E,F)∩K(E,F)

is not closed and the result then follows from Theorem 1.1.

Finally, in [16], the authors attained a general criterium. They introduced the notion of σ-reproducible space in order to obtain the spaceability of the set I1(E, F)\ I2(E, F), where I1 and I2 are general operator ideals in the sense of Pietsch, E or F belongs to that class of spaces and I1(E, F) is complete with the usual operator norm or with another complete ideal norm.

Definition 3.8

A Banach space E is said to be σ-reproducible if there exists a sequence (En)n∈ ℕ of complemented subspaces, where Pn : E ⟶ En is a bounded projection, such that each En is isomorphic to E, Pi∘ Pj = 0 if i ≠ j, and for all k∈ ℕ the projections Pk~=∑n=1kPn:E⟶⊕n=1kEn are uniformly bounded.

Notice that this notion is an isomorphic property. Also, if E and F are σ-reproducible Banach spaces, then E ⊕ F and the dual E* are also σ-reproducible (see [16, Proposition 3.2]).

Every Banach space E with a Schauder decomposition (En)n∈ ℕ (see [23, page 47]) such that each subspace En is isomorphic to E, is σ-reproducible. However, any indecomposable space is not σ-reproducible (recall that a Banach space E is indecomposable if there do not exist infinite-dimensional closed subspaces F and G of E with E = F ⊕ G).

Proposition 3.9

([16]). Every rearrangement invariant space E is σ-reproducible.

Proof

First, let E be a rearrangement invariant space on [0, 1]. For every a ∈ [0, 1) and r ∈ (0, 1 – a] we consider the complemented subspace Ea,r = {x ∈ E : supp x ⊆ [a, a + r]} and the bounded projection Pa,r : E ⟶ Ea,r given by Pa,r(x) = xχ[a, a + r] for x ∈ E.

For a measurable function x we define the linear operators

Ta,r(x)(t)=x(t−ar)χ(a,a+r](t)

and

Sa,r(x)(t)=x((1−t)a+t(a+r))

which are bounded from L∞ to L∞ and from L1 to L1. Thus, using the Calderón-Mitjagin interpolation theorem ([21, Theorem 2.a.10]), Ta, r and Sa, r are bounded from E to E.

These operators also have the following properties:

  1. (Sa,r ∘ Ta,r)(x) = x for every x ∈ E.

  2. (Ta,r ∘ Sa,r)(x) = x for every x ∈ Ea,r.

  3. Ta,r : E ⟶ Ea,r is an isomorphism.

  4. Sa,r : Ea,r ⟶ E is an isomorphism.

Let us show only that Ta,r is injective (it is easy to prove the rest of the properties). Indeed, if x ∈ E \{0}, then there exists n∈ ℕ such that λ(An)>0 where An is defined as An = {t ∈[0,1] : |x(t)| > 1/n}. Thus λ (An′) >0 with An′={(1−t)a+t(a+r):t∈An}. And | Ta,r(x)(s) | > 1/n for every s ∈ An′.

For every n ∈ ℕ we consider an=1−12n−1andrn=12n. Let En = Ean, rn and Pn = Pan, rn. Since ∥Pk~∥=1 for all k ∈ ℕ, we conclude that E is σ-reproducible.

We consider now a rearrangement invariant space E on [0,∞). Let {An : n∈ ℕ} be a disjoint sequence of subsets of [0,∞) where An=⋃k=1∞(an,k,an,k+1] for an increasing sequence (an,k)k∈ ℕ ⊂ ℕ, and the complemented subspaces En = {x ∈ E : supp x ⊆ An}.

Given a measurable function x, we define the linear operators

Tn(x)(t)=∑k=1∞x(t+k−1−an,k)χ(an,k,an,k+1](t)

and

Sn(x)(t)=∑k=1∞x(t+an,k−k−1)χ(k−1,k](t).

Since (Tn(x))* = x*, (Sn (x))*≤x* and (Sn ∘ Tn)(x) = x, we have that Tn : E ⟶ En is an isometry and Sn : En ⟶ E is an isomorphism. Now, reasoning as in the [0, 1] case we obtain the result.

Finally, we consider a symmetric sequence space. Let {Ak : k ∈ ℕ} be a disjoint partition of ℕ where the subset Ak is the range of an injective map k : ℕ ⟶ ℕ for every φk ∈ ℕ.

For x = (xn)n∈ ℕ we define the linear operators Tk(x) = (an)n∈ ℕ with

an=xmifφk(m)=n0ifn∉Ak

and Sk(x) = (xφk(n))n∈ ℕ.

If Ek = {x ∈ E : supp x ⊆ Ak}, we have that Tk : E ⟶ Ek is an isometry and Sk : Ek ⟶ E is an isomorphism. And reasoning again as in the [0, 1] case we obtain the result.

Also, the space C[0, 1 ] is σ-reproducible (see [16, Proposition 3.4]). As a consequence, the space C(K) is σ- reproducible for any uncountable compact metric space K (see [24, Theorem 4.4.8]).

The following lemma will be useful in the proof of the general criterium:

Lemma 3.10

([16]). Let I be an operator ideal.

  1. If E1 and E2 are isomorphic Banach spaces, then there exists a bijection between I(E1, F) and I(E2, F) for every Banach space F.

  2. If G is a closed subspace of the Banach space E and T ∈ I(E, F), then the restriction T |G∈ I(G, F) for every Banach space F.

Theorem 3.11

([16]). Let I1 and I2 be operator ideals such that I1(E, F)\ I2(E, F) is non-empty for a couple of Banach spaces E and F. If E or F is σ-reproducible and I1(E, F) is complete for an ideal norm, then I1(E, F)\ I2(E, F) is spaceable.

Proof

Let T ∈ I1(E, F)\ I2(E, F). If E is a σ-reproducible Banach space with isomorphisms ϕn : En ⟶ E and bounded projections Pn : E ⟶ En, for every n∈ ℕ we consider the operator Tn = T ∘ϕn ∘ Pn which belongs to I1(E, F)\ I2(E, F) . Indeed, using Lemma 3.10, if Tn∈ I2(E, F), then we have that Tn|En = T ∘ϕn ∈ I2(En, F), but this is not true. The sequence (Tn)n∈ ℕ is formed by linearly independent operators. To show this, if ∑n=1k an Tn = 0, restricting to Ej we obtain aj = 0 with 1 ≤ j ≤ k. In the same way, it can be showed that ∑n=1k an Tn cannot belong to I2(E, F) . Thus, I1(E, F)\ I2(E, F) is lineable.

Furthermore, (Tn)n∈ ℕ is a basic sequence in I1(E, F). Indeed, for any integers k < m and any choice of scalars (λn)n∈ ℕ we have

∑n=1kλnTnI1=∑n=1mλnTn∘Pk~I1≤∑n=1mλnTnI1Pk~.

Let S ∈ [Tn:n∈N]¯ ⊂ I1(E, F) with S=∑n=1∞λnTn≠0. Then there exists n0 ∈ ℕ such that λn0 ≠ 0. We have that S | En0 = λn0T ∘ϕn0 ∉ I2(En0, F). Thus, S ∉ I2(E, F) and [Tn:n∈N]¯ ⊂ I1(E, F)\ I2(E, F).

If F is σ-reproducible with isomorphisms (ϕn)n∈ ℕ, for each n∈ ℕ we consider the operator Tn=ϕn−1∘T which belongs to I1(E, F)\ I2(E, F) . The sequence (Tn)n∈ ℕ is formed by linearly independent operators. Thus, we obtain that I1(E, F)\ I2(E, F) is lineable.

And (Tn)n∈ ℕ is a basic sequence. Indeed, for any integers k < m and any choice of scalars (λn)n∈ ℕ we have

∑n=1kλnTnI1=Pk~∘∑n=1mλnTnI1≤Pk~∑n=1mλnTnI1.

Let S ∈ [Tn:n∈N]¯ ⊂ I1(E, F) with S=∑n=1∞λnTn≠0. There exists n0 ∈ ℕ such that λn0 ≠ 0. If S ∈ I2(E, F), then Pn0 ∘ S ∈ I2(E, F), but this is not true because Pn0 ∘ S = λn0 Tn0. Then [Tn:n∈N]¯ ⊂ I1(E, F)\ I2(E, F).

The general criterium can be extended in the following way:

Theorem 3.12

([16]). If E or F is a σ-reproducible Banach space, I is an operator ideal such that I(E, F) is complete for an ideal norm, and (In)n∈ ℕ is a sequence of operator ideals such that the set I(E, F)\ In (E, F) is non-empty for every n∈ ℕ, then the set I(E, F)\ ⋃n=1∞ In(E, F) is spaceable.

Proof

Let Sn ∈ I(E, F)\ In(E, F) for every n∈ ℕ.

If E is a σ-reproducible Banach space with isomorphisms (ϕn)n∈ ℕ and bounded projections (Pn)n∈ ℕ, let us consider the operators Sn ∘ϕn ∘ Pn ∈ I(E, F)\ In (E, F) for every n∈ ℕ. Then

T=∑n=1∞Sn∘ϕn∘Pn2n||Sn∘ϕn∘Pn∥I

belongs to I(E, F)\ In (E, F) for every n∈ ℕ.

Now, reasoning as in the proof of Theorem 3.11 we can construct a sequence (Tk)k∈ ℕ such that [Tk:k∈N]¯ ⊂ I(E, F)\ In (E, F) for every n∈ ℕ.

If F is a σ-reproducible Banach space with isomorphisms (ϕn)n∈ ℕ, let us consider the operators ϕn−1 ∘ Sn ∈ I(E, F)\ In (E, F) for every n∈ ℕ. Then

T=∑n=1∞ϕn−1∘Sn2n||ϕn−1∘Sn∥I

belongs to I(E, F)\ ⋃n=1∞ In(E, F).

Corollary 3.13

([16]). Let E and F be Banach spaces, and {Ip : p ∈[a, b]} be a family of operator ideals such that Ip(E, F) ⊊ Iq (E, F) if p < q with continuous inclusion. If E or F is a σ-reproducible Banach space and Ib(E, F) is complete for an ideal norm, then the set Ib(E, F)\ ⋃p<b Ip(E, F) is spaceable.

In general, Theorem 3.11 does not hold for arbitrary Banach spaces. Consider for instance the spaces with few operators given in [25, 26]. They are hereditary indecomposable Banach spaces on which every bounded linear operator is a compact perturbation of a scalar multiple of the identity.

Finally, let us remark that in the special case of considering Banach operator ideals I1 and I2 with proper continuous inclusion I2⊂ I1 where the ideal I2 is not closed in I1, the set of operators I1(E, F)\ I2(E, F) is always spaceable. This follows from Theorem 1.1.

Many applications of the general criterium can be found in [16] where a good number of particular operator ideals are considered: compact operators, strictly singular operators, strictly co-singular operators, finitely strictly singular operators, (q, p)-summing operators and Schatten operator classes.

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Received: 2017-1-12
Accepted: 2017-2-9
Published Online: 2017-3-20

© 2017 Sánchez

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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https://doi.org/10.1515/math-2017-0019
Keywords for this article
Spaceability; Operator ideal
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